Math 22 Differentiation

The Constant Rule

 The derivative of a constant function is zero. That is, ${\frac {d}{dx}}[c]=0$ where 
 $c$ is a constant

Example: Find derivative of

1) $f(x)=5$

Solution:
$f'(x)=0$

2) $f(x)=\pi$

Solution:
$f'(x)=0$

3) $f(x)=e^{2}$

Solution:
$f'(x)=0$

The Power Rule

 ${\frac {d}{dx}}[x^{n}]=nx^{n-1}$ for $n$ is a real number.

Example: Find derivative of

1) $f(x)=x^{5}$

Solution:
$f'(x)=(5)x^{5-1}=5x^{4}$

2) $f(x)=x^{1000}$

Solution:
$f'(x)=(1000)x^{1000-1}=1000x^{999}$

3) $f(x)={\frac {1}{x^{3}}}$

Solution:
We rewrite $f(x)={\frac {1}{x^{3}}}=x^{-3}$, so
$f'(x)=(-3)x^{-3-1}=-3x^{-4}={\frac {-3}{x^{4}}}$

The Constant Multiple Rule

 If $f$ is a differentiable function of $x$ and $c$ is a real 
 number, then ${\frac {d}{dx}}[cf(x)]=cf'(x)$ for $c$ is a constant.

1) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)=10x^{5}}

Solution:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(x)=10{\frac {d}{dx}}(x^{5})=10(5x^{4})=50x^{4}}

2) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)=3x^{1000}}

Solution:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f'(x)=3{\frac {d}{dx}}(x^{1}000)=3(1000)x^{1000-1}=3000x^{999}}

The Sum and Difference Rules

 The derivative of the sum or difference of two differentiable functions is the sum or difference 
 of their derivatives.
 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {d}{dx}}[f(x)+g(x)]=f'(x)+g'(x)}

 
 ${\frac {d}{dx}}[f(x)-g(x)]=f'(x)-g'(x)$

Notes

1) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {d}{dx}}[x]=1} since Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x^{0}=1}

2) The derivative of $f$ is a function that gives the slope of the graph of $f$ at a point $(x,f(x))$.

3) We usually denote Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {d}{dx}}f(x)} as $f'(x)$

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